Let P be a feasible set without any degenerated vertices Ple

Let P be a feasible set without any degenerated vertices. Please explain that when we work with this P, once we pick the pivot column, there is no freedom for picking pivot row.

Solution

Because we need to avoid cycling. Therefore we have only one freedom:

1) We can select a pivot column among all the given columns.

or 2) We can select a pivot row among all the given rows.

There are mostly used anti-cycling rules:

a) Lexicographic pivoting rules

b) smallest subscript pivoting rule.

Using these two rules, our algorithm does not have cycle and therefore it terminates after a finite or definite number of iterations.